Discovery of energy landscapes towards optimized quantum transport: Environmental effects and long-range tunneling

TL;DR

This work addresses designing energy landscapes to maximize quantum transport in open quasi-one-dimensional chains under environmental effects. It introduces a gradient-based inverse-design approach using Optimistic Gradient Ascent and AdaMax within Lindblad quantum master equation dynamics, examining both local dephasing and finite-temperature baths across short- and long-range tunneling. The main contributions are the identification of distinct landscape classes (nearly flat for short-range coherent transport, corrugated for long-range coherent transport, ramp-like profiles at finite temperature, and ENAQT-enabled regimes) and the extension of these patterns from three-site systems to nine- and ten-site chains, along with mechanistic interpretation via steady-state populations and coherences. The results provide actionable design principles for efficient transport in electronic, photovoltaic, and quantum communication systems, illustrating how energy landscapes can be engineered to harness coherence and environmental effects for robust end-to-end transfer.

Abstract

Carrier transport in quantum networks is governed by a variety of factors, including network dimensionality and connectivity, on-site energies, couplings between sites and whether they are short- or long-range, leakage processes, and environmental effects. In this work, we identify classes of quasi-one-dimensional chains with energy profiles that optimize carrier transport under such influences. Specifically, we optimize on-site energies using Optax's optimistic gradient descent and AdaMax algorithms, enabled by the JAX automatic differentiation framework. Focusing on nonequilibrium steady-state transport, we study the system's behavior under combined unitary and nonunitary (dephasing and dissipative) effects using the Lindblad quantum master equation. After validating our optimization scheme on short chains, we extend the study to larger systems where we identify systematic patterns in energy profiles. Our analysis reveals that different types of energy landscapes enhance transport, depending on whether inter-site tunneling couplings in the chain are short- or long-range, the existence of environmental interactions, and the temperature of the environment. Our classification and insights of optimal energy landscapes offer guidance for designing efficient transport systems for electronic, photovoltaic and quantum communication applications.

Figures (22)

• Figure 1: Scheme of the model: we study transport through a chain of $N$ sites, with tunneling elements $J_{|n-m|}$ between sites $n$ and $m$. Carriers enter site 1 and leave at site $N$ at a rate $\gamma_l$. Local dephasing is enacted on each site (with a rate constant $\Gamma$ in OQS Model I), indicated here with red semitransparent circle on each site. Given tunneling elements, leakage rate, and the dephasing rate, we search for the set $\varepsilon_{1}$ to $\varepsilon_N$ that maximizes the population flux from site $N$.
• Figure 2: Coherent Model: population flux in the absence of environmental effects. Left, (a)-(c): 3-site model with nearest-neighbor tunneling, $J_1=0.2$ and $J_2=0$. (a) Flux map as a function of $\varepsilon_2$ and $\varepsilon_3$, fixing $\varepsilon_1=0$. (b) Level diagram of the optimal configuration, $\varepsilon_2 \in [-3.0, 3.0] \cdot 10^{-4}$ (these values yield the same flux) and $\varepsilon_3 = 1.6 \cdot 10^{-11}$, with the optimal flux $\eta = 0.032$. (c) Steady-state population under the optimal profile, $\rho_{11}^{SS} = \rho_{22}^{SS} = 0.34$, $\rho_{33}^{SS} = 0.32$. Right, (d)-(e): 3-site model with beyond-nearest-neighbor tunneling, $J_1 = 0.2$ and $J_2 = 0.1$. (d) Population flux map. (e) Level diagram of the optimal configuration, $\varepsilon_2 = -0.200$ and $\varepsilon_3 = -0.050$ leading to $\eta = 0.033$. (f) Steady-state population under the optimal profile, $\rho_{11}^{SS} = 0.37$, $\rho_{22}^{SS} = 0.30$, $\rho_{33}^{SS} = 0.33$. Other parameters are $\Gamma = 0$, leakage rate constant $\gamma_l = 0.1$. In panels (a) and (d), results of the OGA algorithm are marked by a black x.
• Figure 3: OQS Model I: population flux in the presence of environmental effects with $\Gamma=0.1$. Left, (a)-(c): model with nearest-neighbor tunneling, $J_1 =0.2$ and $J_2=0$. (a) Population flux map. (b) Level diagram of the optimal configuration, $\varepsilon_2= 1.3 \cdot 10^{-5}$, $\varepsilon_3= 5.8 \cdot 10^{-6}$, resulting in $\eta=0.029$. (c) Steady-state population under the optimal profile, $\rho_{11}^{SS} = 0.37$, $\rho_{22}^{SS} = 0.34$, $\rho_{33}^{SS} = 0.29$. Right, (d)-(e): 3-site model with next-nearest-neighbor tunneling, $J_1 = 0.2$ and $J_2 = 0.1$. (d) Population flux map. (e) Level diagram of the optimal configuration, $\varepsilon_2= -0.292$, $\varepsilon_3= -0.017$, leading to $\eta=0.030$. (f) Steady-state population under the optimal profile, $\rho_{11}^{SS} = 0.36$, $\rho_{22}^{SS} = 0.34$, $\rho_{33}^{SS} = 0.30$. We set $\gamma_l = 0.1$. In both (a) and (d), optimization results of the OGA algorithm are marked by a black x.
• Figure 4: OQS Model II: population flux in the presence of environmental effects at finite temperature, $\Gamma_0=0.1$, $T=0.2$. Left, (a)-(c): 3-site model with nearest-neighbor tunneling, $J_1 =0.2$ and $J_2=0$. (a) Population flux map. (b) Level diagram of the optimal configuration, $\varepsilon_2= 0.21$, $\varepsilon_3= -0.034$, leading to $\eta=0.035$. (c) Steady-state population of the optimal profile, $\rho_{11}^{SS} = 0.40$, $\rho_{22}^{SS} = 0.25$, $\rho_{33}^{SS} = 0.35$. Right, (d)-(e): 3-site model with next-nearest-neighbor tunneling, $J_1 = 0.2$ and $J_2 = 0.1$. (d) Population flux map. (e) Level diagram of the optimal energy profile, obtained at $\varepsilon_2 = -0.118$, $\varepsilon_3 = -0.106$, and leading to $\eta = 0.033$. (f) Steady-state population of the optimal profile, $\rho_{11}^{SS} = 0.36$, $\rho_{22}^{SS} = 0.31$, $\rho_{33}^{SS} = 0.33$. Leak parameter is set at $\gamma_l = 0.1$. In both (a) and (d), the optimized structure from the OGA algorithm is presented by a black x.
• Figure 5: Coherent Model: optimized energy landscape in (a) nine-site and (b) ten-site chains without environmental interactions, for short-range ($\alpha=3$) and long-range ($\alpha=1$) tunneling. (a) $N=9$ sites profile with flux $\eta_{\alpha=3} = 0.0109$ and $\eta_{\alpha=1} = 0.0181$. (b) $N=10$ sites profile with flux $\eta_{\alpha=3} = 0.0098$ and $\eta_{\alpha=1} = 0.0159$. Other parameters are $J_{max} = 0.2$ and $\gamma_l=0.1$. (c)-(d) Steady-state populations for structures corresponding to (a)-(b). (e)-(f) Absolute values of the steady-state density matrix elements (diagonal removed) for the optimized structures in (a). (g)-(h) Absolute values of the steady-state density matrix elements (diagonal removed) for the optimized structures in (b). Other parameters are $J_{max} = 0.2$ and $\gamma_l=0.1$.